Question

A light ray is incident normally on the face $AB$ of a right-angled glass prism $ABC(\mu =1.50)$ as shown in figure. What is the largest angle $\varphi$ for which the light ray is totally reflected at the surface $AC$ ?

• A. ${\sin }^{-1}\left(\frac{2}{3}\right)$
• B. ${\cos }^{-1}\left(\frac{2}{3}\right)$
• C. ${\sin }^{-1}\left(\frac{3}{4}\right)$
• D. ${\cos }^{-1}\left(\frac{3}{4}\right)$

Answer 1

The correct option is B ${\cos }^{-1}\left(\frac{2}{3}\right)$


Let ${\theta }_c$ be the critical angle for glass.
$\frac{\sin {\theta }_c}{\sin {90}^{\circ }}=\frac{1}{\mu }\Rightarrow \sin {\theta }_c=\frac{1}{1.5}=\frac{2}{3}\Rightarrow {\theta }_c={\sin }^{-1}\left(\frac{2}{3}\right)$
From figure, for total internal reflection, ${90}^{\circ }-\varphi >{\theta }_c$
$\Rightarrow \varphi <{90}^{\circ }-{\theta }_c\Rightarrow \varphi <{90}^{\circ }-{\sin }^{-1}\left(\frac{2}{3}\right)$
$\Rightarrow \varphi <{\cos }^{-1}\left(\frac{2}{3}\right)$

So, the largest angle for which light is totally reflected at the surface is ${\cos }^{-1}\left(\frac{2}{3}\right)$.